# Integral of Arcsin[x]

1. Sep 15, 2005

### Jameson

I just can't see it. I would think that it would involve some form of trig substitution, but I'm just drawing a blank. I'll do the work if someone can please give me a nice little hint.

I know that $$\int \arcsin{(x)}dx = \sqrt{1-x^2}+ x\arcsin{(x}) + C$$ from my calculator and mathematica.

2. Sep 15, 2005

### Hurkyl

Staff Emeritus
If only you were working with its derivative! I bet you know how to integrate something that looks like its derivative!

3. Sep 15, 2005

### Jameson

Sure... I'd draw a nice triangle and do trig substitution if it was $$\int \frac{dx}{\sqrt{1-x^2}}$$

Hmmm...I'll think out loud here. If the integral has a square-root and is in the form of $c^2-x^2$ then x is one leg, c is the hypotenuse, and the other leg is the previously mentioned radical.

Is trig substitution with right triangles on the right track? Since the hypotenuse is $\sqrt{a^2+b^2}$, it seems that one leg might need to be $\sqrt{\arcsin{(x)}}$ Somehow I don't think I'm on track.

4. Sep 15, 2005

### amcavoy

Use integration by parts. Remember to let u=arcsin(x) and v=x.

u sub: InverseLogAlgebraicTrigExp

5. Sep 15, 2005

### Hurkyl

Staff Emeritus
Sheesh, just give him the answer, why don't ya? :grumpy:

6. Sep 15, 2005

### Jameson

Ah, thank you! :surprised It's so simple.

And thank you Hurkyl as well. I still have lots to learn.

7. Sep 16, 2005

### amcavoy

I apologize Hurkyl

8. Sep 16, 2005

### tongos

well, you know the integral of sinx with limits. Now arcsin x will be the limits, and you can make a rectangle.

9. Sep 16, 2005

### MalleusScientiarum

Or you could just take the derivative of the right hand side and go "ta da!" and that's proof enough for me.

10. Sep 17, 2005

### professorlucky

hint

equate the arc sine to another variable e.g y.making it a sine fxn.e.g the arc sine of 0.5=30,while sine30=0.5.this will simplify the integral and further substitution will conclude it

11. Sep 19, 2005

### Tide

Just for the fun of it ...

The sum of the integrals

$$\int \sin^{-1} x dx + \int \sin y dy$$

is just the area of the bounding rectangle: $x \times \sin^{-1} x$

Since $$\int \sin y dy = -\cos y + C$$ and $$\cos y = \cos \sin^{-1} x = \sqrt {1-x^2}$$ it follows that
$$\int \sin^{-1} x dx = \sqrt {1-x^2} + x \sin^{-1} x + C$$

12. Jun 9, 2007

### FlashStorm

Integration by parts?

Hey, Im really sorry to arise dead threads from the past (which i have seen though google) but somehting really weird happened me when I tried to use integration by parts on arcsinx.

let me show you:
S(arcsinx)= {v'(x)=1} {u(x)=arcsinx}

xarcsinx-S(x*d(arcsinx))=

xarcsinx-S(x/(1-x^2)^0.5= {u(x)=x v'(x)=arcsinx}

xarcsinx-xarcsinx+S(arcsin)dx

==> S(arcsinx)=S(arcsinx)

:\
I know I have done something really stupid here, but please be easy on me since I started studying Integrals only three days ago.

In the second time I used integration by parts, do I miss something , is there another efficient choice of v and u?

Aviv

p.s: I will edit it better to use normal math signs once I figure out how.

13. Jun 9, 2007

### Gib Z

For the second time you integrate by parts, swap your choices.

14. Jun 9, 2007

### FlashStorm

Im sorry, Tried it also and all i got is:

xarcsine x -(x^2/2)(1/(Sqrt(1-x^2)))-(x/4)(sqrt(1/(1-x^2))+1/4(arcsinx)

this isn't going anywhere :(

15. Jun 9, 2007

### dextercioby

Make $x=\sin t$. Then apply part integration on the resulting integral. It's just a way to avoid the simple solution of part integrating directly.

16. Jun 9, 2007

### Office_Shredder

Staff Emeritus
After the first integration by parts, I would use a substitution

17. Jun 9, 2007

### FlashStorm

solve it officially. Fixed during some major mistakes I had about dev' and stuff.

did it without subtition, only using integration by parts.

if you are interested what I did then you are welcome to tell me to write my solution.

Thanks guys :)
gg

18. Oct 10, 2008

### josiahsavino

use seperation by parts

u=arcsinx
du=dx/(1-x^2)^1/2

dv=dx
v=x

uv-ingegral vdu = xarsinx-integral x/(1-x^2)^1/2
use u substitution with u = 1-x^2 so du = -2x
than you get

xarsinx + (1-x^2)^1/2 + c

19. Apr 27, 2010